Is there a way to play all actions that are part of one armature one after another?
You need to push down each action that you want to show from the Dope Sheet into the NLA (Nonlinear Animation window):

Then open the NLA and move each strip (you can also put them all on the same track). Press G to move a strip, right click on a track to move it up or down:

If you press N in the NLA you'll have access to the NLA properties, for example you can decide the effect of the selected strip, Extrapolation &gt; Hold means that the strip below the one selected won't be able to act. If you want them to act, choose Nothing, or put the strips above this one:

If you want the NLA to play you need to close the action playing in the Dope Sheet, action playing in the Dope Sheet will play primarily.
To quickly edit an action, select its strip in the NLA and press Tab, the strip will turn green, the action will automatically appear in the Dope Sheet.